Hyperspatial optimization of structures

Anticipating the low-energy arrangements of atoms in space is an indispensable scientific task. Modern stochastic approaches to searching for these configurations depend on the optimization of structures to nearby local minima in the energy landscape. In many cases these local minima are relatively high in energy, and inspection reveals that they are trapped, tangled, or otherwise frustrated in their descent to a lower-energy configuration. Strategies have been developed which attempt to overcome these traps, such as classical and quantum annealing, basin/minima hopping, evolutionary algorithms, and swarm-based methods. Random structure search makes no attempt to avoid the local minima and benefits from a broad and uncorrelated sampling of configuration space. It has been particularly successful in the first-principles prediction of unexpected new phases of dense matter. Here it is demonstrated that by starting the structural optimizations in a higher-dimensional space, or hyperspace, many of the traps can be avoided and that the probability of reaching low-energy configurations is much enhanced. Excursions into the extra dimensions are progressively eliminated through the application of a growing energetic penalty. This approach is tested on hard cases for random search: clusters, compounds, and covalently bonded networks. The improvements observed are most dramatic for the most difficult ones. Random structure search is shown to be typically accelerated by two orders of magnitude and more for particularly challenging systems. This increase in performance is expected to benefit all approaches to structure prediction that rely on the local optimization of stochastically generated structures.

Figure 1

Flowcharts for the geometry optimization of structures from hyperspace (GOSH), random structure search (RSS) using GOSH, and relax and shake (RASH) using GOSH.

Figure 2

Defining the distance $l_i$ of atom $i$ in hyperspace to normal space for $d=d_0+d_{+}=1+2$.

Figure 3

A Lennard-Jones trimer for $d=1+1$. Two identical Lennard-Jones 4-2 atoms (with $\sigma =1)$ are fixed at $(\pm 1,0)$, and a third atom seeks its lowest energy. Steepest-descent paths from (i) (2.75,0), constrained to $d_0$ (black), (ii) $(-2.75,1)$ in $d=1+1$ (white), and (iii) $(-2.75,-1)$ in $d=1+1$ with increasing $\mu$ (red).

Figure 4

Density of structural states for 38-atom Lennard-Jones clusters (inset: 55 atoms, $f=0.1)$. Snapshots of an optimization of a random initial structure with $d_{+}=1$, whose local minimum corresponds to the $O_h$ nonicosahedral ground state, are presented. The atoms are color-coded according to their location in the extra fourth dimension: increasingly red for positive values and blue for negative values. The probability of encountering the ground state is provided in parentheses as a fraction $\left(6/10\mathrm{K}=6/10000\right)$. Note that for high packing densities with $d_{+}=0$ the $O_h$ ground state is generated with high probability, and so the peak in the structural density of states is expected to shift farther leftwards with increasing $f$.

Figure 5

The optimization of an initially random binary structure packed into a three-dimensional hypersphere, with $d=1+2$, toward the ground state in $d_0=1$.

Figure 6

Density of states for a model 1D binary system, with 12 atoms of type A (red, “positive”) and 13 atoms of type B (blue, “negative”). Random structures were generated with $f=0.3$. The gaps between fragments have been reduced for presentational purposes.

Figure 7

Evolution of the encounter probability with system size for the one-dimensional $\left(d_0=1\right)$ binary Lennard-Jones system. The gray dashed line indicates the probability of randomly choosing the correct ordering.

Figure 8

Density of states for a model 3D binary system, with 13 atoms of type A (red) and 14 atoms of type B (blue) and a low packing fraction of $f=0.05$ for the initial random structures.

Figure 9

Local minimum of a model system with defined connectivity and $d_{+}=0$. Random initial configurations typically relax into topologically frustrated configurations.

Figure 10

Model networked system. The packing fraction of the initial random structures is $f=0.1$. A sequence of snapshots for an optimization of a structure whose local minimum corresponds to a low-energy sheet for $d_{+}=1$ is presented. See Fig. 11 for further details.

Figure 11

Snapshots of a geometry optimization of a model system with defined connectivity and $d_{+}=1$. The atoms are colored according to their position in the extra dimension: red for positive values and blue for negative ones. The locally optimal atoms rapidly leave hyperspace. Where the network remains tangled, the atoms continue to inhabit the extra dimension and appear to move through each other in normal space. See Ref. [53] for an animation of the optimization.